Question: Find one value of $x$ that is a solution to the equation: $(2x-3)^2=4x-6$ $x=$
We could solve for $x$ by expanding $(2x-3)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a shorter and more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $4x-6=2({2x-3})$. This means that we can rewrite the equation as: $({2x-3})^2=2({2x-3})$ If we let ${p}={2x-3}$, we can see that this equation is in the form: ${p}^2=2{p}$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2&=2{p}\\\\ {p}^2-2{p}&=0\\\\ {p}({p}-2)&=0\\\\ {p}=0\ &\text{or} \ \ {p}=2 \end{aligned}$ Since ${p}={2x-3}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${2x-3}=0\ \ \ \text{or} \ \ \ {2x-3}=2$ When we solve ${2x-3}=0$, we find that $x=\dfrac{3}{2}$. When we solve ${2x-3}=2$, we find that $x=\dfrac{5}{2}$. In conclusion, the two solutions of the equation $(2x-3)^2=4x-6$ are $x=\dfrac{3}{2}$ and $x=\dfrac{5}{2}$. [Is there another way to solve for x?]